On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that {\em contracts edges} is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting $k$ edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the \textsc{$\cal F$-Contraction} problem, where $\cal F$ is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph $G$ and an integer $k$, \textsc{ $\cal F$-Contraction} asks whether there exists $X \subseteq E(G)$ such that $G/X \in \cal F$ and $|X| \leq k$. Here, $G/X$ is the graph obtained from $G$ by contracting edges in $X$. We obtain the following results for the \textsc{ $\cal F$-Contraction} problem. $(1)$ We show that \textsc{Clique Contraction} admits a polynomial-size approximate kernelization scheme (\textsf{PSAKS}). $(2)$ We give a $(2+ε)$-approximate polynomial kernel for \textsc{Split Contraction} (which also implies a factor $(2+ε)$-\FPT-approximation algorithm for \textsc{ Split Contraction}). Furthermore, we show that, assuming \textsf{ Gap-ETH}, there is no $\left(\frac{5}{4}-δ\right)$-\FPT-approximation algorithm for \textsc{Split Contraction}. Here, $ε, δ>0$ are fixed constants. $(3)$ \textsc{Chordal Contraction} is known to be \WTH. We complement this result by observing that the existing \textsf{W[2]-hardness} reduction can be adapted to show that, assuming \FPT $\neq$ \textsf{W[1]}, there is no $F(k)$-\FPT-approximation algorithm for \textsc{Chordal Contraction}. Here, $F(k)$ is an arbitrary function depending on $k$ alone.